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1. Unique Decoding of Explicit -balanced Codes Near the Gilbert-Varshamov Bound

   https://doi.org/10.1109/FOCS46700.2020.00048  

   Jeronimo, Fernando Granha; Quintana, Dylan; Srivastava, Shashank; Tulsiani, Madhur (November 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS))

   null (Ed.)

   The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance 1/2-ε and rate Ω(ε 2 ) (where an upper bound of O(ε 2 log(1/ε)) is known). Ta-Shma [STOC 2017] gave an explicit construction of ε-balanced binary codes, where any two distinct codewords are at a distance between 1/2-ε/2 and 1/2+ε/2, achieving a near optimal rate of Ω(ε 2+β ), where β→ 0 as ε→ 0. We develop unique and list decoding algorithms for (a slight modification of) the family of codes constructed by Ta-Shma, in the adversarial error model. We prove the following results for ε-balanced codes with block length N and rate Ω(ε 2+β ) in this family: -For all , there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time N Oε,β(1) . -For any fixed constant β independent of ε, there is an explicit construction of codes which can be uniquely decoded up to an error of half the minimum distance in time (log(1/ε)) O(1) ·N Oβ(1) . -For any , there are explicit ε-balanced codes with rate Ω(ε 2+β ) which can be list decoded up to error 1/2-ε ' in time N Oε,ε' ,β(1), where ε ' ,β→ 0 as ε→ 0. The starting point of our algorithms is the framework for list decoding direct-sum codes develop in Alev et al. [SODA 2020], which uses the Sum-of-Squares SDP hierarchy. The rates obtained there were quasipolynomial in ε. Here, we show how to overcome the far from optimal rates of this framework obtaining unique decoding algorithms for explicit binary codes of near optimal rate. These codes are based on simple modifications of Ta-Shma's construction. 

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2. Minimum L_∞ Hausdorff distance of point sets under translation: Generalizing Klee's measure problem

   https://doi.org/10.4230/LIPIcs.SoCG.2023.24  

   Chan, Timothy M. (June 2023, Proc. 39th Sympos. Computational Geometry (SoCG))

   Chambers, Erin W.; Gudmundsson, Joachim (Ed.)

   We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis. 

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3. A Lower Bound for Sampling Disjoint Sets

   https://doi.org/10.1145/3404858  

   Göös, Mika; Watson, Thomas (July 2020, ACM Transactions on Computation Theory)

   Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [ n ] and Bob ends up with a set y ⊆ [ n ], such that ( x , y ) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω ( n ) communication even to get within statistical distance 1− β n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Ω (√ n ) communication is required to get within some constant statistical distance ɛ > 0 of the uniform distribution over all pairs of disjoint sets of size √ n . 

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4. A Lower Bound for Sampling Disjoint Sets

   https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.51  

   Göös, Mika; Watson, Thomas (January 2019, Proceedings of the 23rd International Conference on Randomization and Computation (RANDOM))

   Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [n] and Bob ends up with a set y ⊆ [n], such that (x, y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω(n) communication even to get within statistical distance 1 − β^n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Ω(√n) communication is required to get within some constant statistical distance ε > 0 of the uniform distribution over all pairs of disjoint sets of size √n. 

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5. A Distanced Matching Game, Decremental {APSP} in Expanders, and Faster Deterministic Algorithms for Graph Cut Problems

   https://doi.org/10.1137/1.9781611977554.ch82  

   Chuzhoy, Julia (January 2023, Proceedings of the 2023 {ACM-SIAM} Symposium on Discrete Algorithms,{SODA} 2023)

   Expander graphs play a central role in graph theory and algorithms. With a number of powerful algorithmic tools developed around them, such as the Cut-Matching game, expander pruning, expander decomposition, and algorithms for decremental All-Pairs Shortest Paths (APSP) in expanders, to name just a few, the use of expanders in the design of graph algorithms has become ubiquitous. Specific applications of interest to us are fast deterministic algorithms for cut problems in static graphs, and algorithms for dynamic distance-based graph problems, such as APSP. Unfortunately, the use of expanders in these settings incurs a number of drawbacks. For example, the best currently known algorithm for decremental APSP in constant-degree expanders can only achieve a (log n) O(1/ 2 ) -approximation with n 1+O( ) total update time for any . All currently known algorithms for the Cut Player in the Cut-Matching game are either randomized, or provide rather weak guarantees: expansion 1/(log n) 1/ with running time n 1+O( ) . This, in turn, leads to somewhat weak algorithmic guarantees for several central cut problems: the best current almost linear time deterministic algorithms for Sparsest Cut, Lowest Conductance Cut, and Balanced Cut can only achieve approximation factor (log n) ω(1). Lastly, when relying on expanders in distancebased problems, such as dynamic APSP, via current methods, it seems inevitable that one has to settle for approximation factors that are at least Ω(log n). In contrast, we do not have any negative results that rule out a factor-5 approximation with near-linear total update time. In this paper we propose the use of well-connected graphs, and introduce a new algorithmic toolkit for such graphs that, in a sense, mirrors the above mentioned algorithmic tools for expanders. One of these new tools is the Distanced Matching game, an analogue of the Cut-Matching game for well-connected graphs. We demonstrate the power of these new tools by obtaining better results for several of the problems mentioned above. First, we design an algorithm for decremental APSP in expanders with significantly better guarantees: in a constant-degree expander, the algorithm achieves (log n) 1+o(1)-approximation, with total update time n 1+o(1). We also obtain a deterministic algorithm for the Cut Player in the Cut-Matching game that achieves expansion 1 (log n) 5+o(1) in time n 1+o(1), deterministic almost linear-time algorithms for Sparsest Cut, Lowest-Conductance Cut, and Minimum Balanced Cut with approximation factors O(poly log n), as well as improved deterministic algorithm for Expander Decomposition. We believe that the use of well-connected graphs instead of expanders in various dynamic distance-based problems (such as APSP in general graphs) has the potential of providing much stronger guarantees, since we are no longer necessarily restricted to superlogarithmic approximation factors. 

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